Talk:Polygon

AxelBoldt removed the statement:

Strictly speaking, every polyhedron is also a polygon as is every polytope, since they all have angles.

with the simple claim:

Polyhedra are not polygons.

Since it is obvious that polyhedron have multiple angles, and hence are polygonal, I'd like to give him a chance to explain why he removed true information.

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Simple: because it's not true at all. "Polygon" is almost universally defined as a 2-dimensional figure. I don't know of any mathematics text or course that treats it as a superclass that includes 3-d polyhedra. Terms here should be used as they are commonly used in academia. --Lee Daniel Crocker

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Lee, perhaps you would like to suggest what term should be used for the class of objects that have multiple angles, regardless of the dimentionality? Then we could put in a reference to that class of objects in this article. I had never heard the term polytope before I got involved with Wikipedia. Does the concept of the angle between two planes make sense? I, admittedly, have found very little accessible material on these topics. -- BenBaker

Maybe a phrase such as:

Even though strictly speaking, every polyhedron has multiple angles, as does every polytope, they are not considered as polygons as the angles between their faces are not two dimensional. They can be classified as 'technical-term', however.

"Polytope" is the general term, although it is typically only used to refer to 4-d and higher figures (because the 2- and 3-d figures already have names). It is, nonetheless, proper to refer to polygons and polyhedra as subclasses of polytopes. --LDC

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Mathematical terms are not defined etymologically. "Polygon" may mean "many angles" in Greek, but that doesn't mean that anything with many angles is called a polygon in mathematics (and yes, you can have angles between planes). Polygons are two-dimensional figures that enclose an area with straight lines. We could have links to polytopes and polyhedra I suppose. --AxelBoldt

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I am not aware of any word in any context that is defined by its etymology. Words mean whatever they are defined to mean, regardless of where they happen to come from. Adding an explicit statement to that effect in this article would be silly, because that's just a case of understanding the nature of the English language and has nothing to do with polygons. This article is about polygons, which are flat. Now, if you want to add some statement to the effect that polygons are the two-dimensional instance of the more general class of polytopes, that's entirely appropriate. --LDC

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"Words mean whatever they are defined to mean, regardless of where they happen to come from." -- Indeed. That's the Humpty Dumpty argument! (Through the Looking-Glass)

I'd like to mention the term "n-gon" on this page, since that's the link I followed to get here. Also, the table is somewhat inconsistent: a "Triangle" may be regular or not. A "Square" is regular by definition. The other terms usually are taken to mean the regular form -- in my experience it's more common to see a phrase like "an irregular pentagon" than "a regular pentagon".

Fixed. --Damian Yerrick

On the dimension issue, it might be fair to mention that a polygon is a 2D polytope, but it's not terribly interesting. The question of a "broken" polygon in higher dimensions -- ie a set of non-planar points joined by a closed, simple path -- is perhaps interesting, but completely breaks the definition of a polytope as a convex hull of point, and there's no longer any notion of area or volume. I suppose then it's merely a path. -- Tarquin

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Hi. I added a proposed taxonomy, but it does have problems. There is the problem that under the definitions that I left, a complex polygon may be considered convex. Is this indeed the case? If so, the two versions of convex are surely nevetheless considered distinct, so Simple convex and complex convex are distinct classes, both denoted 'convex'? Or am I just being too hopeful in proposing a tree-based taxonomy?